Concepts · Number series

Landmark numbers

Some numbers you should know on sight, the way you know your own street.

Riya has just learned the difference table, and she likes it. She meets this:

121, 144, 169, ?, 225

Gaps: 23, 25. So the next gap is 27, and the blank is 196; then 225 − 196 = 29 fits too: gaps 23, 25, 27, 29, and their gaps 2, 2, 2. It works. It also takes her a minute, and she has a feeling it should not have.

The next question is worse:

2, 3, 5, 7, 11, 13, ?

Gaps: 1, 2, 2, 4, 2. Gaps of the gaps: 1, 0, 2, −2. Their gaps: −1, 2, −4. Nothing settles. The table that saved her five minutes ago has nothing to say.

Left at the temple

That evening Kabir is coming over for the first time, and he rings from the main road.

“How do I find your house?”
“Past the petrol pump, left at the temple, third house on the right.”
“How many houses is that from where I am?”
“No idea.”

She puts the phone down and thinks about it. She has never counted those houses. You find your way by the few places you know on sight, and count only the last few steps from there. Her two series were built from places like that: 121 and 144 are squares, and 2, 3, 5, 7 are primes. Numbers have landmarks too.

The rule

Know these on sight: the squares from 1² to 25², the cubes from 1³ to 10³, the powers of 2 up to 2¹⁰, the factorials from 1! to 7!, and the 25 primes up to 100.

Here they all are. This concept is more memory than method, and the squares are the place to start. (A factorial n! means 1 × 2 × … × n, so 4! = 1 × 2 × 3 × 4 = 24.)

121, 144, 169 are 11², 12², 13². The blank is 14² = 196: one glance, no subtraction.

Riya’s question, again

2, 3, 5, 7, 11, 13 are the primes, in order. The gaps between primes follow no simple rule, which is exactly why the table found nothing. No simple pattern in the gaps finds the next prime. The list does, or checking each number for factors: 14 = 2 × 7, 15 = 3 × 5 and 16 = 2 × 8 can each be split into smaller factors, so none of them is prime, and the next prime is 17.

Landmarks in disguise

A landmark can hide behind a small shift: one more, or one less. 2, 9, 28, 65, ? looks like nothing, until you check each term against the nearest landmark: 1, 8, 27, 64 are cubes, and every term is one more. The next cube is 125, so the next term is 126.

So when a series looks almost familiar, check each term against the nearest landmark, and see how far each term is from it.

Why it helps in the paper

A square series can be solved slowly with a table, and a prime series not at all. A student who knows the landmarks reads the same line as a list of old friends.

Series that are built by adding: The difference table.

Try three

Next mock

A week later: 50, 65, 82, ? Riya does not write a single gap. 49, 64, 81 are squares, and each term is one more. The next square is 100. She writes 101.